Near-thermo-neutral electron recombination of titanium oxide ions

where A and B symbolize the atomic fragments. As an exothermic non-radiative capture process, the DR reaction (1) can have a very high rate in cold gas-phase ionized media and by far dominates over the omnipresent radiative electron–ion recombination. Being a fundamental chemical process, DR has motivated a wide range of experimental and theoretical investigations. Often, these studies revealed intriguing beyond-Born–Oppenheimer dynamics behind the dissociation process, making the full understanding of DR a still challenging task.

A class of transition metal oxides (MOs) have been identified to potentially violate the exothermic DR behavior.³ For some MO cations, DR toward ground-state atomic fragments can become endothermic, implying that any dissociative electron capture process requires a net energy input into the system. Compared to the usual exothermic DR case, MO ions with endothermic DR are expected to much less efficiently neutralize with surrounding electrons.

Previous studies based on chemical thermodynamic data³ identified a set of transition and lanthanide metal elements whose oxide ions MO⁺ potentially display the endothermic DR behavior. Notably, this group of elements includes titanium, which is also known to have a significant cosmic abundance.⁴ Thus, studying the DR of TiO⁺ on the one hand addresses endothermic DR as a fundamental process, while, on the other hand, it can also be expected to yield new evidence for the chemistry of titanium in cosmic environments.

The role of titanium atoms and titanium oxide molecules in astrophysical media has been discussed for the last five decades (e.g., Tsuji, Ref. 5). The TiO and TiO₂ observations in hot stellar and circumstellar environments⁶ suggest that these molecules are essential in the dust grain formation process, acting as seeds for further silicate growth.⁷ In the colder interstellar medium (ISM), atomic titanium is known to be strongly depleted.^8–10 Thus, also there, titanium oxide molecules, such as TiO, TiO₂, and, potentially, even TiO⁺, are expected to behave as titanium storage. However, none of the searches for titanium oxide molecules in the ISM have been successful so far.^11–13 

As discussed further in Sec. II B, the pre-existing molecular and atomic data suggest that reaction (3) is close to thermoneutral, within an uncertainty of several tens of millielectronvolt (few hundred kelvin equivalent). More precise data are thus needed to finally resolve the question of CI exothermicity or, reciprocally, DR endothermicity. The new data may greatly improve our understanding on the titanium ISM chemistry.

To the best of our knowledge, neither the CI of Ti + O nor the DR of TiO⁺ has been investigated with respect to their reaction dynamics in previous experimental or theoretical work. The understanding of the corresponding energy balance is currently limited by the accuracy of TiO⁺ and TiO dissociation energies derived from room-temperature chemical reaction studies (Sec. II B). In this type of study, better energy resolution is hindered by the broad rotational excitation distribution of the molecules.

In our study, we investigate the energy balance of TiO⁺ DR using the Cryogenic Storage Ring (CSR) at the Max Planck Institute for Nuclear Physics in Heidelberg, Germany. In the low-temperature radiation field of this setup, the TiO⁺ molecules first cool down to the lowest rotational levels. Then, the energy balance of TiO⁺ is probed by collisions with cold electrons at controlled collision energy while measuring the energy output of the reaction in the form of kinetic energy released to Ti and O fragmentation products. Additionally, we determine collision-energy dependent rate coefficients for the TiO⁺ DR at various levels of TiO⁺ internal excitation.

The remainder of this paper is structured as follows: In Sec. II, we discuss the energy balance of the TiO⁺ DR reaction with a special emphasis on the molecular structure of TiO⁺. Section III describes the CSR facility and the applied experimental approach. Section IV discusses the data analysis and presents the derived TiO⁺ DR reaction energy and the reaction rate coefficients. New quantum chemical calculations on the energy balance in the TiO⁺ DR and on TiO potential energy curves are presented in Sec. V. Finally, in Sec. VI, the experimental and calculated results are compared and discussed in the context of molecular dynamics theory and astrochemistry.

where $ETiO+$⁠, E_Ti, and E_O are the internal excitation energies of the involved reactants and products; E_c is the electron–ion collision energy in the center-of-mass frame; E_KER is the kinetic energy released to the Ti and O products, i.e., the center-of-mass kinetic energy of the products in their final states; and ΔE is the reaction energy for the ground-state reactants. It is here defined as the DR endothermicity (strictly speaking, endoergicity), i.e., as the minimum collision energy needed for the reaction to proceed from the ground state TiO⁺ to produce Ti and O in their respective ground states. Thus, positive ΔE suppresses the DR of ground-state TiO⁺ ions with electrons in the limit of zero collision energy.

Figure 1 shows the essential energy structure of the TiO and TiO⁺ molecules (potential curves calculated by Miliordos and Mavridis, Ref. 15) and the energy levels of the Ti and O reaction products. Given the low, sub-eV energies involved in our experimental approach, the states most relevant to our study lie energetically close to the TiO⁺(X²Δ) ground state. This includes the electronic fine-structure splitting and ro-vibrationally excited states of TiO⁺ and also the electronic states of Ti and O. These are the asymptotic, dissociated energy levels of neutral TiO electronic potential curves, whose detailed correlations to the separated-atom levels, however, are beyond the scope of this work.

The TiO⁺(X²Δ) ground term features a fine-structure doublet splitting¹⁶ of (26.26 ± 0.05) meV [spin–orbit coupling constant A = (105.9 ± 0.2) cm⁻¹], leading to X²Δ_3/2 and X²Δ_5/2 states with projection quantum numbers of the total angular momentum along the internuclear axis of Ω = 3/2 and 5/2, respectively. Higher electronically excited states A²Σ⁺, B²Π, and a⁴Δ appear only above ∼1.4 eV, as listed in Table I. The vibrational level spacing for both X²Δ_3/2 and X²Δ_5/2 is¹⁶  $∼0.13$ eV.

Both atomic products of TiO⁺ DR display triplet fine-structure splitting in the ground terms O(³P) and Ti(a³F) with individual excitation energies of $<50$ meV. The lowest electronic product states are summarized in Table I. The total internal energy levels of both fragments, E_Ti + E_O, relevant for the E_KER analysis according to Eq. (5), are shown in Fig. 1.

To determine the reaction energy and the collision energy-dependent rate coefficient of the TiO⁺ DR reaction, we use the merged electron–ion beam technique employed in the electrostatic Cryogenic Storage Ring (CSR)²⁴ facility at the Max Planck Institute for Nuclear Physics in Heidelberg, Germany. Our study partially follows the approach used in the past for other DR studies at the CSR, e.g., for HeH⁺, CH⁺, and OH⁺ ions.^25–27 Therefore, only a brief description of the general experimental approach is given here, complemented by the specific details on the TiO⁺ DR project.

We generated an ∼20 nA beam of TiO⁺ ions using a sputtering Penning ion source (see  Appendix A). The beam was accelerated to an energy of E_ion = 280 keV, passed through a magnetic mass filter (set to a mass-to-charge ratio of m/q = 64 u/e), and then, about 10⁶ ions were injected into the CSR. Once in the storage ring [Fig. 2(a)], the TiO⁺ ion beam was circulated at a frequency of $∼26$ kHz around the $∼35$ m closed orbit of the ring for a defined storage time. Due to the cryo-pumping effect, the temperature of $<6$ K in the CSR chamber results in residual gas densities of only $∼103$ cm⁻³, allowing for the TiO⁺ storage of up to $∼1600$ s (exponential beam lifetime $∼1000$ s). As discussed in more detail in  Appendix B, the low radiation field in CSR allows the ions to radiatively decay closer to their electronic and rovibrational ground states. This reduction in the internal excitation energy spread is beneficial for the DR energy balance measurements (lowering of E_KER ambiguities). After the radiative cooling phase, which extended for up to 1500 s, the DR reaction was probed by merging the stored ion beam with a collinear electron beam. The neutral DR products (Ti and O) were collected by using a microchannel plate (MCP) detector in the cryogenic region. The entire procedure (an injection cycle) was repeated until sufficient statistical quality of the data was obtained.

The low energy electron beam for the DR measurements was achieved using the electron cooler device. The electrons were extracted from a Ga–As photocathode, and the resulting beam was accelerated and expanded in the magnetic field to achieve low energy spreads of k_BT_⊥ ≈ 2 meV and k_BT_‖ = 0.2–2.5 meV in the transverse and longitudinal directions, respectively. Here, k_B stands for the Boltzmann constant. Guided by a solenoidal magnetic field, the electron beam was co-linearly merged with the ion beam in an electron–ion interaction zone.

In the DR-probing phase of the injection cycle, the electron detuning energy E_d was swapped repeatedly between four values with a dwell time of few tens of milliseconds each. In the first step, E_d is set to 0 eV (i.e., E_e = E₀). In this step, the electron beam drag force guarantees that the ion beam energy is kept constant even though non-zero detuning energies are applied in the other steps.³¹ In the next step, the energy is detuned to a desired value for DR measurement E_m. Here, E_m varies between the swapping cycles so that all desired collision energies are covered. In the next step, E_d was fixed to a reference value E_d = E_r ≈ 0.2 eV. The neutral-fragment count rate during this step was used as a proxy for the relative ion beam intensity, as further explained in  Appendix D. In the last step, the electron beam was switched off to collect background data from processes not induced by electrons (among others, residual gas collisions and the detector dark count rate). Within each injection cycle, the four-step sequence was repeated until the duration of the DR probing phase was covered. A 5 ms waiting time was added between each of the steps to allow the electron beam to stabilize after changing the acceleration conditions.

The neutral products (Ti and O) resulting from DR in the electron interaction zone and from the ion collisions with the residual gas were separated from the stored ion beam in the downstream electrostatic deflector of the storage ring lattice. The products propagated ballistically to a 120 mm-diameter MCP detector backed by a phosphor (P-screen) anode.³² Here, the fragment impact positions manifested themselves as rapidly decaying light spots on the P-screen. Optical readout by using a silicon photomultiplier was used to provide triggers for impact-event counting. From the event count rates, the DR rate coefficient was derived as further discussed in Sec. III C. The time separation of the fragment impacts from a single DR event was well below 1 µs and was thus resolved as single trigger only. At the typical impact rate of several kHz, deadtime effects and mixing of DR events within a single trigger can be neglected.

A separate, $∼1$ kHz frame-rate camera was used to read out the impact positions of DR fragments event by event with a spatial accuracy of ∼0.1 mm. The distance between the fragments reflects the kinetic energy release E_KER as further discussed in Sec. III B.

We aim to deduce the TiO⁺ DR reaction energy ΔE from the kinetic energy release E_KER using the energy balance of Eq. (5). In turn, E_KER is determined from the Ti and O product distances as measured at the MCP-based imaging detector. A large advantage of our experimental method is that the collision energy E_c occurring in Eq. (5) can be varied. Hence, also such DR channels that are energetically forbidden because of positive ΔE or because of the fragment excitation energies E_Ti and E_O can be observed. Correspondingly, E_KER can be tracked by controlled variations of E_c, realized by varying the beam detuning energy E_d.

Measured transversal fragment distributions and their comparison to the analytical $f⊥0(d;EKER)$ were often used in DR experiments^2,33,36 to determine E_KER for single isolated fragmentation channels, branching ratios, internal ionic excitation, and angular fragment distributions. However, in the present case, a large number of fragment channels with densely spaced E_KER are expected to overlap. They represent, in particular, the nine possible DR channels corresponding to combinations of fine structure excitation in Ti and O (see Table I with I_Ti = 0…2 and I_O = 0…2) and to the fine-structure and rotational excitation of TiO⁺ remaining even at long storage times (see  Appendix B). All these contributions superimpose on each other with different amplitudes that cannot be predicted theoretically. In this situation, it is still practical for a general picture to consider the peak positions d_peak of possibly contributing DR channels in relation to the measured fragment distance distributions $f⊥̃(d)$⁠. However, in order to obtain information about the underlying reaction energy ΔE, a more elaborate analysis is required. We model experimental transverse fragment distributions as a superposition of all possible DR channels in a detailed approach described in  Appendix C. With relative DR channel amplitudes and the reaction energy ΔE as parameters, this yields model distributions f_⊥(d; ΔE) used to derive ΔE in a least-squares fitting procedure. In particular, this procedure aims at obtaining the value of ΔE that best represents the ensemble of experimental distributions $f⊥̃(d)$ observed for a range of detuning energies E_d between the ion and electron beams.

For our experimental parameters, Eq. (11) can be numerically written as $EKER=dpeak2×4.821$ meV/mm² using D_min = 3300 mm, E_ion = 280 keV, and the Ti and O isotopic masses. This yields d_peak = 4.55 mm (7.89 mm) for E_KER = 100 meV (300 meV). At the typical electron energy spread represented by k_BT_⊥ ≈ 2 meV, well-defined variations of E_c can be envisaged in steps down to $∼10$ meV. Hence, observing corresponding variations of the fragment distance distributions on the mm scale potentially offers access to the reaction energy balance [Eq. (5)] with an accuracy down to a few meV.

An example of measured fragment distance distributions $f⊥̃(d)$ is given in Fig. 4 for velocity-matched beam conditions (E_d = 0) and several storage-time intervals with DR sampling. While the distributions clearly show the strong decrease in fragment distances as the ions internally deexcite with increasing storage time (further discussed in Sec. IV A), the data also reflect the limited position resolution of the detector and include minor background contributions.

The fragment positions have been acquired by using a CMOS camera in an optical configuration with $∼2.4$ pixels corresponding to 1 mm in the detector plane. Each of the impact-generated spots on the phosphor screen anode covered an area of $∼20$ square-pixels following a Gaussian-like 2D distribution. Using the light intensities from the individual pixels, the spot-center position is determined with a precision of $∼0.1$ mm. However, spots from impacts at low transverse distances can overlap and thus affect the spot-position evaluation procedure. We find that the $f⊥̃(d)$ distribution is undistorted only for distances at d ≥ d_min = 1.6 mm. Hence, we also limit the fitting by the model distribution to distances d ≥ d_min. In the plots of acquired $f⊥̃(d)$ (e.g., Fig. 4), we also include the data in the range of d = 1.2–1.6 mm. Here, only a small distortion is expected, while the overall distribution shape can still be used for qualitative discussion. With Eq. (11), the limit d_peak = d_min corresponds to E_KER ∼ 12 meV.

Acquired fragment-pair events originate not only from the DR of TiO⁺ but also contain a background from collisions of the ions with the residual gas, from dark counts of the detector, and from DR of the TiOH⁺ contaminant (see  Appendix A). The first two background contributions can easily be corrected for by subtracting reference data acquired without electron beam. Data in all $f⊥̃(d)$ plots in this paper have been corrected in this way. The contribution from TiOH⁺ DR cannot be subtracted as easily. Instead, we include the TiOH⁺ contribution in the model fragment distance distribution f_⊥(d; ΔE) as a term $f⊥bg(d)$⁠, as described in  Appendix C.

The TiO⁺ DR rate coefficient measurement has been performed by using the electron beam as a collision target for the stored TiO⁺ ions. By adjusting the duration of the radiative cooling phase prior to the DR probing phase used in all injection cycles, the DR rate coefficient could be determined for various levels of TiO⁺ internal excitation.

The experimentally determined DR rate coefficient as a function of the detuning energy E_d is given as the merged-beams rate coefficient α^mb(E_d), obtained from the ion beam current, electron beam density, detector counting efficiency, and further experimental parameters, as explained in  Appendix D. The rate coefficient α^mb reflects the electron–ion collision energy distribution specific to the particular CSR experiment. We present α^mb(E_d) for different storage-time intervals, which represent the various TiO⁺ internal excitation conditions realized in our experiment.

We also converted each of these results α^mb(E_d) to a kinetic-temperature rate coefficient α^k(T^k), which assumes a Maxwell–Boltzmann distribution at variable kinetic temperature T^k for the collision energies. The conversion procedure is described in  Appendix E. In the literature, the kinetic-temperature rate coefficient α^k is often denoted as “plasma rate coefficient.” Note that, in this procedure, the TiO⁺ internal excitation is independent of T^k and continues to be given by the excitation conditions in the CSR for the respective storage time window.

We acquired fragment distance distributions $f⊥̃(d)$ at various DR-probing storage-time windows, ranging from 0 to 1600 s. The data were acquired over several injection cycles for a set of detuning energies E_d = 0, 10, 20, 30, 40, 60, 90, 120, and 150 meV. The evolution of the distance distribution $f⊥̃(d)$ as a function of storage time is represented in Fig. 4 for E_d = 0 eV. For determining the reaction energy ΔE, using the complete set of E_d values, we aimed at high signal intensity in connection with a good degree of TiO⁺ internal relaxation and, therefore, focused on the storage-time window of 500–600 s. The results for $f⊥̃(d)$ in this storage time window and for the set of non-zero E_d are represented in Fig. 5. In all displayed datasets, non-DR contributions have been subtracted already so that the remaining signal represents the DR of TiO⁺, partly contaminated by the signal from DR of TiOH⁺.

The E_d = 0 eV data in Fig. 4 demonstrate the internal cooling of TiO⁺ ions when stored in the CSR. In the 0–20 s probing interval, the $f⊥̃(d)$ distribution peaks at d ≈ 6 mm with a decaying tail at even larger distances. As d_peak = 4.55 mm for E_KER = 100 meV from Eq. (11), this shows that, even though E_d = 0 eV, the fragment energies exceed 100 meV for the hot ions soon after their injection into the CSR. At later storage times, the fragment distances strongly decrease and the $f⊥̃(d)$ distribution peaks at distances well below d = 1.6 mm (E_KER < 12 meV for d_peak < 1.6 mm). Similarly, for all storage times $≥500$ s, only a decaying tail up to d ≲ 3 mm is found (E_KER = 43 meV for d_peak = 3 mm). As discussed in  Appendix C, the remaining flat part of $f⊥̃(d)$ at d > 3 mm can be well assigned to the DR of TiOH⁺. Hence, we conclude that for the TiO⁺ DR, a strong reduction of E_KER occurs after longer storage.

Following the energy balance equation [Eq. (5)], we rationalize this observation by the internal cooling of the TiO⁺ ions. This is also supported by the fact that the drop of the observed E_KER follows well the trend of predicted TiO⁺ internal excitation from our radiative cooling model [see Fig. 2(b) and Table III]. Alternatively, the lower E_KER at long storage times could originate from increased excitation of the product Ti and O atoms (see Table IV) for internally cold TiO⁺. However, such a strong dependence of product channel branching ratios on internal excitation has never been observed theoretically or experimentally.

Considering the energy balance in Eq. (5), the internal relaxation implies a decrease of the term $ETiO+$⁠, where an average $⟨ETiO+⟩=11±2$ meV is expected in the storage-time window of 500–600 s according to  Appendix B. Hence, the observed strong storage-related decrease of E_KER at E_d = 0 eV suggests that the sum E_Ti + E_O + ΔE is not strongly negative and can hardly be smaller than −20 meV. In this balance, however, the relative contributions of DR channels with various values of E_Ti and E_O remain unknown. In particular, it cannot simply be assumed that the DR signal remaining at E_d = 0 eV and low $⟨ETiO+⟩$ is due to ground-state Ti and O products only. To probe the participation of excited Ti and O products, we analyze the measurements at non-zero E_d, where all these DR channels (at least for the nine combinations of the lowest Ti and O fine-structure levels) are expected to become gradually accessible. While the individual amplitudes of these channels still remain uncertain, it can be expected that the fragment distance distributions for higher E_d will to some degree reflect the participation of all possible fine-structure excited DR channels.

According to  Appendix C, Table IV, the values of E_Ti + E_O span a range of 76 meV with an irregular pattern, roughly appearing in four groups of more closely spaced values. In E_KER from Eq. (5), this leads to similarly distributed values, where in addition the energies $ETiO+$ from the various rotational levels (J,Ω) of TiO⁺ will be added. In the full distribution f_⊥(d), the respective transverse distance distributions for the various E_KER will be summed up.

In view of the complex situation of unknown and E_d-dependent relative contributions of the individual DR channels, we will proceed in two steps. First, we allocate a range with the limits ΔE_min and ΔE_max in which ΔE can at most vary considering the experimental $f⊥̃(d)$⁠. Here, we compare the observed fragment distance distributions to the peak positions d_peak expected for single-E_KER channels from Eq. (11). In a second step, we vary ΔE within this range to obtain modeled transverse distance distributions f_⊥(d, ΔE) [ Appendix C, Eq. (C1)] where the individual channel amplitudes are determined from independent least-squares fits of f_⊥(d, ΔE) to $f⊥̃(d)$ for the various E_d values probed. From the overall minimum of the mean squared deviations χ² reached for the different ΔE, we extract a most likely value of ΔE and estimate its uncertainty.

In the first step, we compare the acquired $f⊥̃(d)$ with a stick diagram f_SD(d), which represents the values d_peak from Eq. (11). Here, E_KER is obtained from the energy balance equation [Eq. (5)] for each possible channel combination (J, Ω, I_Ti, I_O) and setting E_c = E_d. This yields f_SD(d) as a sum of δ-like-function peaks at d = d_peak, over all the channels, with finite peak amplitudes weighted by the rotational-level populations p_J;Ω from the radiative cooling model ( Appendix B). The relative weights between the different (Ω, I_Ti, I_O) channels are arbitrarily set to 1. The resulting stick pattern of f_SD(d) is finally positioned along d by setting the reaction energy ΔE in Eq. (5) and compared with experimental $f⊥̃(d)$ for different ΔE.

Figure 6 shows the comparison for two values of E_d (rows) and four values of ΔE (columns). The experimental $f⊥̃(d)$ used in the upper row (E_d = 10 meV) shows contributions from TiO⁺ DR up to $∼2.5$ mm. The stick diagrams for different ΔE show that, irrespective of the individual peak amplitudes in the stick diagram, such contributions exist only if the reaction energy is below 40 meV. As a conservative upper limit, we hence obtain ΔE_max = 40 meV.

Conversely, the experimental $f⊥̃(d)$ in the lower row of Fig. 6 (E_d = 150 meV) shows a peak position near 4.6 mm, showing that DR channels exist with E_KER down to the value corresponding to this d_peak. If the reaction energy would be smaller than −10 meV, DR channels with such a small E_KER would not exist, irrespective of the individual channel amplitudes. Hence, as a lower limit, we obtain ΔE_min = −10 meV. Similar behavior compatible with these upper and lower limits of ΔE is found for the data at the other E_d.

In the second step, we use the detailed transverse fragment distance model f_⊥(d) of Eq. (C1) to fit the experimental data $f⊥̃(d)$⁠, varying the channel amplitudes of $AITi;IO;Ω$ and the background amplitude B, i.e., all free parameters except of ΔE. Details of the fitting procedure are given in  Appendix C 3. With ΔE at a fixed value between ΔE_min and ΔE_max, the mean squared deviations χ²(ΔE, E_d) are combined to a reduced value $χred2(ΔE)$ according to Eq. (C3). The results for a grid of ΔE values within the range allocated in the first step are shown in Fig. 7.

The overall $χred2(ΔE)$ (red circle symbols) is seen to assume a minimum at ΔE ≈ +4 meV, where $χred2(ΔE)≈1.5$⁠. Toward lower ΔE, the function $χred2(ΔE)$ displays a smooth rise without any additional features. Toward higher reaction energies, a localized increase of $χred2(ΔE)$ occurs at ΔE ≈ 15 meV, followed by a secondary minimum at ΔE ≈ 24 meV. In Fig. 7, we also include the reduced squared deviations from only the E_d = 10 meV dataset, $χred,10meV2(ΔE)$ (blue triangle symbols). At this very small E_d, the model distributions f_⊥(d, ΔE) turn out to be particularly sensitive to ΔE, and for ΔE > 0, a good fit to $f⊥̃(d)$ (see also Fig. 12) is only possible when ΔE lies close to the overall minimum near 4 meV, while the model for ΔE near the secondary minimum (24 meV) clearly cannot explain the experimental data, as also reflected by the large values of $χred,10meV2(ΔE)$ in this region. The specific function $χred,10meV2(ΔE)$ shows another minimum for ΔE < 0. However, this minimum contradicts the data with higher E_d, taken into account in the full $χred2(ΔE)$⁠. In fact, at small E_d, the sensitivity of f_⊥(d, ΔE) to the specific values of ΔE is lost for ΔE < 0 since many combinations of channel amplitudes can explain the experimental shape in this case. We note that the $χred,Ed2(ΔE)$ curves for E_d other than 10 meV display shallow broad structures in the discussed ΔE region and thus do not contradict our reasoning here.

We acquired the DR data for TiO⁺ by probing the ion beam by electrons at various storage time windows, thus accessing different levels of TiO⁺ internal excitation. For each storage time window, the merged-beams rate coefficient α^mb was derived as a function of detuning energy E_d, following Eq. (D2) derived in  Appendix D. Averaged within DR-probing windows of 0–20 s, 60–120 s, 500–600 s, 1000–1100 s, and 1500–1600 s, respectively, the resulting rate coefficients α^mb(E_d) are shown in Fig. 8. One-sigma error bars reflecting the uncertainty by the counting statistics are included in Fig. 8, while the total systematic scaling uncertainty was determined to be 34%, as described in  Appendix D.

The α^mb rate coefficient is dominated by DR of ⁴⁸Ti¹⁶O⁺, complemented by other minor components. At detuning energies E_d ≳ 6.9 eV (corresponding to the TiO⁺ binding energy²⁰), the dissociative excitation (DE) of TiO⁺ by electrons becomes energetically accessible, resulting in Ti⁺ and O fragments. The applied detection technique cannot distinguish between the O fragments, reaching the detector after a DE event and the pair of Ti and O fragments resulting from DR. Therefore, the rate coefficient increase at E_d ≳ 6.9 eV may be, at least in part, given by the DE. On the other hand, for most diatomic molecular ions, DR is enhanced at E_d ≳ 10 eV due to the opening of new pathways for the reaction via higher lying repulsive (unbound) neutral potential curves.¹ While such potential curves, converging to ground-state products Ti(a³F) + O(³P), can indeed be expected, we are unaware of any experimental or theoretical data that would allow us to safely assign the rate at E_d > 10 eV to either DR or DE.

Another contribution to α^mb is expected to arise from the DR of TiOH⁺. The stored ⁴⁸Ti¹⁶O⁺ ion beam was partly contaminated by ⁴⁷Ti¹⁶OH⁺ isobars ( Appendix A). In  Appendix C 2, we show that TiO⁺ DR and TiOH⁺ DR can be almost completely distinguished from each other via their strongly different E_KER and, consequently, different transverse distances of the DR products on the detector. For analyzing the contribution from TiOH⁺ DR, we choose the 500–600 s dataset and the collision energy range of E_d ≤ 150 meV. At the given storage times, the TiO⁺ ions are strongly deexcited and the transverse distances of Ti and O fragments from TiO⁺ DR do not exceed d = 8 mm for all E_d ≤ 150 meV. On the other hand, the large E_KER available in the DR of TiOH⁺ results in two-fragment events covering the full area of the 120 mm-diameter detector. Thus, to estimate the relative TiOH⁺ DR signal contribution, we compare the rate of two-fragment imaging events with d > 8 mm to the total two-fragment signal. Additionally, we take into account the fact that part of the two-fragment events cannot be resolved due to overlapping spots on the P-screen of the detector (d ≲ 1.6 mm) or escape registration because of too large transverse distance, in which case the lighter O fragment can miss the active detector area. We also consider various shapes of the TiOH⁺ DR signal at d < 8 mm [$f⊥bg(d)$⁠, discussed in  Appendix C 2]. The uncertainty of these effects is propagated to the final estimate of the TiOH⁺ DR contribution, as displayed by the dark-gray bars in Fig. 8 for detuning energies E_d = 0, 30, 60, 90, 120, and 150 meV. In the 500–600 s window, the fraction of the TiOH⁺ DR signal within the total DR signal ranges from 14% to 26%. We note that the plotted TiOH⁺ data represent the absolute DR rate coefficient of TiOH⁺ scaled by the relative fraction of the TiOH⁺ ions in the stored beam (⁠$<10$%; see  Appendix A).

Based on the comparatively more complex structure of TiOH⁺ with smaller rotational and vibrational energy intervals, we assume that its radiative cooling rate is slow compared to that of TiO⁺, especially at the $>500$ s storage times. Therefore, the TiOH⁺ DR rate estimate derived from the 500–600 s storage-time data should be valid also for the later storage times. To our best knowledge, theoretical or experimental data on the DR of TiOH⁺, which we could use for comparison, are unavailable. In Fig. 8, we add a hypothetical merged-beams rate coefficient assuming cross section varying as $σ(Ec)∝Ec−1$ (representing the most usual DR behavior for polyatomic ions¹) and convolve it according to Eq. (E1) to yield α^mb. This background model clearly shows a steeper collision-energy dependence than the estimated experimental background. A better match was reached by creating a background model with a $σ(Ec)∝Ec−0.85$ cross section dependence, as also indicated in Fig. 8. A more detailed analysis TiOH⁺ background would go beyond the scope of this paper.

The α^mb(E_d) curves reveal strong signal variations, both with E_d and with the storage time. At low detuning energies (E_d ≲ 30 meV, especially for short storage times), the strong decay of the rate coefficient with the increasing energy is a usual DR behavior.^1,2 Additionally, we here observe a strong decrease of the rate coefficient for long storage times, which can be attributed to the radiative deexcitation of TiO⁺ and to the endothermicity of the DR process. As further discussed in Sec. VI, at long storage times, the TiO⁺ DR channel becomes energetically inaccessible for an increasingly large fraction of TiO⁺ ions as they reach low-lying rotational levels.

For the use of our DR data in plasma applications at various kinetic temperatures T^k, we converted the measured merged-beams rate coefficient α^mb(E_d) to the kinetic temperature rate coefficient α^k(T^k) following the procedure given in  Appendix E. Additionally, we subtracted the estimated contribution from the TiOH⁺ DR contaminant signal using the $σ∝Ec−0.85$ cross section in the background model as discussed for α^mb in Fig. 8. The resulting α^k(T^k) curves for the individual storage time windows are plotted in Fig. 9 in a temperature range T^k = 10–10⁴ K. The statistical uncertainties were propagated from α^mb to α^k and are displayed by the colored shaded areas in Fig. 9. They peak at the lowest temperatures of T^k ≈ 10 K and do not exceed 33%. The systematic uncertainty is dominated by the absolute scaling uncertainty (34%) and by uncertainties from the longitudinal and transverse electron temperatures T_‖ and T_⊥, respectively, which contribute via the deconvolution procedure of α^mb to the cross section (see  Appendix E and references therein). The longitudinal temperature k_BT_‖ varies with the laboratory-frame electron energy and ranges from $∼2.4$ meV for E_d = 0 eV to $∼0.2$ meV at E_d ∼ 10 eV. Its uncertainty is dominated by a residual voltage ripple on the cathode potential and propagates to α^k as up to 50%. For the transverse temperature, $kBT⊥=2.0−0.5+1.0$ meV is assumed, which propagates to α^k as a scaling factor of up to 33%. The systematic uncertainties added in quadrature are shown as the additional gray band around the 500–600 s curve in Fig. 9 and reach up to 65% at T^k ≈ 10 K, while dropping down to $∼35%$ for T^k ≳ 300 K. The systematic uncertainties for the other curves are similar and are therefore not shown in Fig. 9. Also not shown in Fig. 9 is the estimated uncertainty from subtracting the TiOH⁺ DR rate, which reaches at most 10% for the 1500–1600 s curve and is lower for the other storage times.

We complement our measurements of the TiO⁺ reaction energy ΔE by state-of-the-art quantum chemical calculations. To derive the DR reaction energy ΔE, i.e., the energy difference between the lowest energy Ti(4s²3d²; ³F₂) + O(³P₂) fragments and the ground vibrational level (v = 0) of TiO⁺(X²Δ_3/2), we perform coupled-cluster calculations with single, double, and perturbatively connected triple excitations, CCSD(T), and apply a correction for spin–orbit splitting. The wavefunctions of both TiO⁺ (at equilibrium) and the atomic fragments (M_L = ±2; ³A₁ under the C_2v point group) are of single-reference nature justifying the use of CCSD(T). The correlation consistent basis sets cc-pVXZ (X = T, Q, 5)^37,38 were employed to construct the atomic and molecular orbitals. A series of diffuse basis functions were added in the basis set of oxygen (aug-cc-pVXZ).³⁹ The equilibrium energies E_e, distances r_e, and harmonic vibrational frequencies ω_e for all basis sets are listed in Table S1 of the supplementary material [equilibrium constants for TiO(X³Δ) are also provided for completeness]. We also included the electron correlation from the sub-valence 3s²3p⁶ electrons of titanium. These core–valence calculations are denoted as CV-CCSD(T), and the corresponding weighted core basis sets were used for titanium (cc-pwCVXZ).³⁷ Scalar relativistic effects were then added with the second order Douglas–Kroll–Hess (DKH2) Hamiltonian. In these calculations, CV-CCSD(T)-DKH2, the appropriate cc-pwCVXZ-DK basis set³⁷ was used for titanium combined with the uncontracted aug-cc-pVXZ sets for oxygen. The equilibrium energies and frequencies were then extrapolated to the complete basis set (CBS) limit, assuming exponential convergence. Finally, spin–orbit corrections were taken from experimental data. Specifically, the energy decrease from Ti(³F) to Ti(³F₂) and from O(³P) to O(³P₂) was estimated as the energy difference between the M_J-averaged energy values of Ti(³F) and O(³P) and their lowest energy J = 2 component. The energy values were taken from Ref. 40, and the final spin–orbit corrections are ΔE_SO(Ti; ³F) = −0.0276 eV and ΔE_SO(O; ³P) = −0.0097 eV. For TiO⁺, we used⁴¹ ΔE_SO(TiO⁺; X²Δ) = −ΔE_e(X²Δ_5/2-X²Δ_3/2)/2 = −0.0260 eV/2 = −0.0130 eV since the two Ω components have the same degeneracy. From these values, ΔE is lower than the energy difference neglecting spin–orbit interaction by 0.0243 eV. Similarly, from TiO spectroscopy,⁴² we find ΔE_SO(TiO; X³Δ) = −ΔE_e(X³Δ₃-X³Δ₁)/2 = −0.0244 eV/2 = −0.0122 eV, which we use to find the spin–orbit corrections for D₀(TiO).

Table II lists ionization energies (IE_e and IE₀) and binding energies (D_e and D₀) for TiO obtained from the energies and frequencies of Table S1, where ω_e/2 was used for the zero-point vibrational corrections for IE₀ and D₀. The values of IE_e and IE₀ are practically identical, considering theoretical uncertainties, differing by less than 0.004 eV at any level of theory. Our best IE₀ value [CV-CCSD(T)-DKH2] is 6.829 eV, which within $∼0.01$ eV is consistent with the experimental value of (6.8198 ± 0.0001) eV¹⁶ and 0.014 eV away from the recent coupled-cluster calculated value²³ of 6.815 eV (including corrections due to quadruple electronic excitations). Note that the relativistic corrections are necessary to get this high level of agreement. Plain CCSD(T) and CV-CCSD(T) give 6.629 and 6.749 eV, respectively, at the CBS limit. On the other hand, the electron correlation energy of the 3s²3p⁶ electron of titanium is important for obtaining a very accurate binding energy. Plain CCSD(T) gives D₀ equal to 6.677 eV, which increases by 0.173 to 6.850 eV at CV-CCSD(T) and drops to 6.827 eV when scalar relativistic effects are included. Finally, the calculated energy differences ΔE between Ti(³F) + O(³P) and TiO⁺(X²Δ; v = 0) fluctuate around 0.0 eV depending on the methods or basis sets used. Positive values mean that TiO⁺ is lower in energy. Small basis sets tend to favor the TiO⁺ level, while at the CBS limit going from CCSD(T) to CV-CCSD(T), ΔE increases (both positive values). On the other hand, going from CV-CCSD(T) to CV-CCSD(T)-DKH2, ΔE drops and turns negative. The spin–orbit corrections, CV-CCSD(T)-DKH2-SO, favor the Ti(³F) + O(³P) level, decreasing ΔE even further to −0.0263 eV. Similarly, spin–orbit corrections to be applied to D₀(TiO) are −0.0251 eV, and the result of 6.827 eV from Table II is changed to 6.802 eV. In Sec. VI, these values will be compared to the present experiment and to results from others. Within that discussion, the calculated quantities will be referred to as ΔE^calc = −26 meV and $D0calc=6.802$ eV.

To identify the crossings between the potential energy curve (PEC) of TiO⁺(X²Δ) and the repulsive part of the TiO PECs, we performed multi-reference configuration interaction (MRCI) calculations with aug-cc-pVTZ basis sets. The PECs in the region of the crossings are shown in Fig. 10. The MRCI calculations are based on complete active space self-consistent field (CASSCF) wavefunctions with the active space composed of the 4s3d/_Ti 2p/_O orbitals. The valence 2s/_O orbital was not included in the active space because of convergence issues, but excitations from it to the virtual space were included at the MRCI level. For the DR reaction, the spin coupling TiO⁺(S = 1/2; X²Δ) + e⁻(S = 1/2) generates only singlets and triplets, and thus, only singlet and triplet states of TiO are considered here. The calculations were done within the C_2v point group, and the TiO states were averaged as ¹A₁/¹A₂ (¹Σ^±,¹Δ,¹Γ), ¹B₁/¹B₂ (¹Π,¹Φ,¹H), ³A₁/³A₂ (³Σ^±,³Δ,³Γ), and ³B₁/³B₂ (³Γ,³Φ,³H) separately at the CASSCF level. The PEC of TiO⁺ was calculated at the state specific CASSCF + MRCI level. To account for the inconsistency between the TiO and TiO⁺ PECs (state-averaged vs state-specific), we shifted the PECs of TiO⁺ by 0.029 698 a.u. (0.808 eV) to match the experimental TiO/TiO⁺ energy difference (rounded value from Ref. 16 of 6.82 eV). Figure 10 also includes the energy and wavefunction of the vibrational ground state of TiO⁺. The TiO ground state fragments Ti(4s²3d²; ³F) + O(³P) generate the ^1,3(Σ⁺[2], Σ⁻[1], Π[3], Δ[3], Φ[2], Γ[1]) states, and nearly all of them cross the ground state TiO⁺(v = 0) PEC within the Franck–Condon region (see Fig. 10). It should be emphasized that the above-mentioned energy shift moves the crossings by only ∼−0.04 Å, and thus, the shifting does not change the main observation that the TiO PECs cross the TiO⁺ PEC in the equilibrium region.

Using the combined merged-beams and fragment-imaging techniques, we experimentally determined the reaction energy for the dissociative recombination of TiO⁺ of ΔE = (+4 ± 10) meV (positive ΔE indicates endothermic DR). The reaction is thus thermoneutral within the uncertainties, with a slight bias toward endothermicity. Our result is well compatible with the value of ΔE^prev = (+0.05 ± 0.07) eV obtained from previous experimental data (Sec. II B), while the uncertainty improved by a factor of 7. In turn, our new value for ΔE can be used to derive updated values for the dissociation energies of TiO and TiO⁺ by inverting Eqs. (7a) and (7b). We obtain D₀(TiO) = ΔE + IE(TiO) = (6.824 ± 0.010) eV and D₀(TiO⁺) = ΔE + IE(Ti) = (6.832 ± 0.010) eV, respectively. Also here, the precision of the new dissociation energies improves by a factor of 7 compared to the best previous experimental values.^19,20

Additionally, we determined values for the reaction energy ΔE^calc and the TiO dissociation energy $D0calc(TiO)$ using state-of-the-art coupled-cluster calculations. Our theoretical result ΔE^calc = −26 meV differs from the experiment by 30 meV—a difference well within the scatter between the various levels of the theory (see Sec. V). Interestingly, previous coupled-cluster calculations on TiO and TiO⁺ by Pan et al.²³ yield ΔE^calc = +33 meV. On the other hand, our calculated $D0calc(TiO)=6.802$ eV lies by 30 meV below the experimental result derived above. Other coupled-cluster calculations²³ yield $D0other(TiO)=6.853$ eV.

Furthermore, we measured the TiO⁺ DR merged-beams rate coefficient α^mb(E_d), which represents the reaction rate as a function of the collision energy at the experiment-specific energy spread (Fig. 8). The rate coefficient displays a decrease by a factor of up to 30 as the TiO⁺ ions deexcite while stored in the CSR. This trend is strongest at small collision energies E_d ≲ 10 meV and continues even at the longest investigated storage times of 1500–1600 s. Given the observed near-thermoneutrality of TiO⁺ DR and the expected low TiO⁺ excitation (Table III), the strong reaction to the internal excitation implies that efficient pathways leading to DR become active as soon as the energetic threshold of the reaction is overcome.

To better understand this, we investigated the PECs of those TiO electronic states, which may be formed by the capture of thermal electrons (down to a few meV) on TiO⁺ and then dissociate to Ti + O reaction products. Generally, in a DR reaction, the electron is captured by the molecular ion to form a neutral molecule in an energy-resonant excited state above the ionization threshold that dissociates into neutral fragments. For the “usual” exothermic DR, the final states can be reached without a barrier even in the limit of the lowest collision energies E_c. For this case, various pathways have been described.^1,2,36 In particular, if neutral dissociating states cross the ionic potential within the Franck–Condon region of the ion, electron capture into these states is possible without any sharp dependence on E_c or on the internal molecular excitation energy (“direct” DR). In contrast, sharp features as a function of E_c are expected from “indirect” DR, where resonant electron capture occurs into ro-vibrationally excited neutral Rydberg states, which then pre-dissociate by coupling with another, dissociating excited neutral state. Barrier-less (exothermic) DR may still be suppressed for excited final states.

Since the process is governed by the electrostatic interaction in the multi-electron system, the excited states relevant for the DR of TiO⁺(X²Δ, v = 0) ions are expected to be those with total spin S = 0 and 1, given the spin 1/2 of the free electron. Accordingly, our calculated singlet and triplet PECs for TiO are shown in Fig. 10 together with the TiO⁺ ground-state PEC. The neutral curves correlate with various levels of the atomic dissociation products within an energy band of $∼0.08$ eV, as shown in Fig. 1. A large number of relevant neutral PECs cross the TiO⁺ ground state within the Franck–Condon region and would give rise to direct DR channels without a sharp energetic dependence in the “usual” case of exothermic DR. In the case of TiO⁺, however, minimum energies δE exist for reaching all or most of the correlated final states. In a complex manner, the exact δE for each channel will depend on the potentials at larger internuclear distance, including possible barriers in the adiabatic potentials, the individual final atomic levels, and also the internal excitation of the colliding TiO⁺ ion. Nevertheless, it is plausible that for each of the channels, the DR rate will turn on with the typical smooth energy dependence of a “direct” DR process once the corresponding threshold δE is reached. Below the threshold, high vibrational levels in any of the electronic states of neutral TiO may resonantly couple to the TiO⁺ + e⁻ collision channel—although these levels cannot dissociate to Ti + O. Instead, they can either re-autoionize to TiO⁺ + e⁻ or stabilize to a lower-lying neutral TiO level by photon emission (photon-stabilized recombination). Small typical rates of only $∼10−12$ cm³ s⁻¹ are generally envisaged² for processes of the latter type.

Experimentally, we do not observe any significant resonant features in our DR spectra. This indicates that indirect DR, driven by neutral Rydberg levels at E_c > δE, is small for this system. Moreover, consistent with their expected small rates, features from resonant radiative recombination are also not observed. Instead, the experimental data for α^mb(E_d) and their dependence on the ion storage time suggest the presence of “direct,” but slightly endothermic DR pathways whose thresholds become more and more effective as the storage time increases and the internal excitation of the TiO⁺ ions reduces.

We demonstrate this by a model for α^mb(E_d) that takes into account the rotational and fine structure excitation of the TiO⁺ ions (see Sec. II A and  Appendix B). Since the relative contributions of fine-structure excited final Ti and O states (Fig. 1) are unknown, we resort to the strongly simplifying assumption that all recombination leads to the Ti and O ground states so that DR fully sets in when the sum of E_c and the TiO⁺ excitation energy $ETiO+(J,Ω)$ exceeds the endothermicity ΔE. Then, neglecting also the relative differences between the DR cross sections of the various channels, we assume the rate coefficient to be proportional to the number of TiO⁺ rotational levels above energetic threshold. For each of these levels, we assume a DR cross section varying as ∝1/E_c for $Ec>ΔE−ETiO+(J,Ω)$⁠, as typical for direct DR.² We then convolve these cross sections with the experimental energy distribution to obtain a model for α^mb(E_d) (see  Appendix F for more details).

After a scaling to the experimental data at E_d > 0.1 eV, the resulting model is shown in Fig. 11 for the experimental storage times and an example DR endothermicity ΔE = +14 meV. At the lowest E_d, the model, similar to the experimental data (Fig. 8), shows a rapid decrease of the rate coefficient as a function of storage time. In particular, at later times, only the remaining population in the Ω = 5/2 branch still contributes to DR. As E_d rises, Ω = 3/2 levels of TiO⁺ also contribute, resulting in the peak at E_d ≈ 20 meV. At even higher E_d, the model α^mb(E_d) decreases, reflecting the ∝1/E_c dependence, while, independent of storage time, all TiO⁺ levels contribute. The experimental data for the 500–600 s time window, included in Fig. 11, have a similar DR rate at low E_d as obtained for the model at ΔE = +14 meV. The same model for ΔE = +4 meV (not plotted) would show significantly higher rates in the same storage time window. This comparison alone could be taken as an indication that, within the uncertainty range of the result from the KER analysis, the larger ΔE values are more likely. However, the comparison in Fig. 11 also reveals a more complicated energy dependence of α^mb(E_d) than predicted by the simple model, together with a shift in the intermediate peak up to E_d ∼ 0.06 eV. This likely reflects the more detailed variation of the DR cross section with energy and between the channels corresponding to the different neutral PECs, as well as the influence of fine-structure excited Ti + O product channels. Reaching a better match between the model and the experimental data by including more details in the DR model is beyond the scope of this work. However, even the presented simple model strongly suggests that the decrease of the rate at low energies is due to the rotational cooling of the TiO⁺ levels below the energetic threshold and that the low-energy TiO⁺ DR rate coefficient would further decrease for even colder TiO⁺ ions.

From our measured α^mb(E_d), we also derived kinetic-temperature rate coefficients α^k(T^k) for the TiO⁺ DR in cold ionized media (Fig. 9) pertaining to the TiO⁺ excitation conditions realized in our various storage time windows. Especially at low temperatures (T^k ≲ 1000 K), the kinetic rate coefficients α^k(T^k) quickly drop as the internal TiO⁺ excitation becomes lower, reflecting the trend in the α^mb(E_d) data. In the dataset for the longest storage times (1500–1600 s), the TiO⁺ ions are expected to have on average $∼3.5$ meV internal excitation, with 98% of the population in the eight lowest rotational levels of the X²Δ_3/2 ground electronic state and $∼2$% remaining in the X²Δ_5/2 levels with a radiative equilibrium not yet reached. In interstellar space (thin, cold ISM), molecules often de-excite down to equilibrium with the cosmic microwave background at $∼2.7$ K or remain at temperatures of the order of 10 K because of collisional heating processes. For TiO⁺ under such equilibrium conditions, the limits of T_exc = 10 K (2.7 K) would imply significant population in only few of the lowest rotational levels for X²Δ_3/2 and very small relative populations of roughly 10⁻¹³ (10⁻⁴⁸) in X²Δ_5/2. Based on the strong influence of the internal excitation of TiO⁺ within our simplified model of the DR process, it can be expected that compared to the experimental data, the rate coefficient would further drop for such even lower excitation stages.

Altogether, we consider our kinetic rate coefficient α^k(T^k) for the longest storage times (1500–1600 s) as an upper limit for cold-ISM conditions, which amounts to α^k ≤ 1 × 10⁻⁷ cm³ s⁻¹ for T^k = 10 K. To estimate the lower limit, we consider the largest endothermicity within our uncertainty range, ΔE_max = 14 meV, where only the X²Δ_5/2 levels could still contribute to DR at the low ISM temperatures. Assuming that the rate coefficient scales with the change of the X²Δ_5/2 level population, the kinetic rate coefficient α^k(T^k) at T^k = 10 K would reach down below α^k < 5 × 10⁻¹³ cm³ s⁻¹ (10⁻⁴⁷ cm³ s⁻¹) for the discussed cold ISM conditions at T_exc = 10 K (2.7 K). Hence, our results include the possibility that the rate coefficient for the recombination of TiO⁺ ions with electrons in the cold ISM at T^k = 10 K would instead of DR be dominated by the non-dissociative radiative recombination with a generally predicted level of $∼10−12$ cm³ s⁻¹. Nevertheless, with the increasing kinetic temperature, the DR channel is expected to rise rapidly and reach a level between 10⁻⁸ and 10⁻⁷ cm³ s⁻¹ at T^k = 100 K, similar to our 1500–1600 s experimental data.

In summary, we demonstrated that, even for a very complex case, the combined merged-beams and fragment-imaging techniques applied on internally cold ions in a cryogenic storage ring can yield thermochemistry data with a 10 meV accuracy. The TiO⁺ recombination with electrons is found to be thermoneutral within this uncertainty. At the cold (⁠$<10$ K) conditions in the interstellar medium, the reaction cannot proceed faster than $∼1×10−7$ cm³ s⁻¹—a rate already an order of magnitude lower than usual for most other diatomic molecular ions. In addition, the low-temperature DR rate coefficient of TiO⁺ may be even significantly lower and is expected to vary strongly with the internal excitation of these ions. If TiO⁺ could ever be detected in cold ISM, its abundance would likely be a highly sensitive probe for the TiO⁺ internal excitation and, hence, the balance between collisional excitation processes and radiative relaxation. In the future, we plan to apply the here presented experimental approach to further systems expected to feature endothermic molecular-ion recombination, such as ZrO⁺.

See the supplementary material for the list of newly calculated Ti, O, and TiO+ energies and equilibrium parameters.

Financial support by the Max Planck Society is acknowledged. A.K. was supported, in part, by the U.S. National Science Foundation Division of Astronomical Sciences Astronomy and Astrophysics Grants program under Grant No. AST-1907188. E.M. acknowledges the James E. Land endowment (Auburn University) and the U.S. National Science Foundation (Grant No. CHE-1940456) for their financial support. AFRL support under Air Force Office of Scientific Research (Grant No. AFOSR-22RVCOR009) is acknowledged. The views expressed are those of the authors and do not reflect the official guidance or position of the Department of the Air Force, the Department of Defense (DoD), or the U.S. government. The appearance of external hyperlinks does not constitute endorsement by the United States DoD of the linked websites, or the information, products, or services contained therein. The DoD does not exercise any editorial, security, or other control over the information you may find at these locations.

The authors have no conflicts to disclose.

Naman Jain: Data curation (equal); Formal analysis (equal); Software (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Ábel Kálosi: Data curation (equal); Formal analysis (equal); Software (equal); Visualization (equal); Writing – review & editing (equal). Felix Nuesslein: Conceptualization (equal); Investigation (equal); Methodology (equal); Writing – review & editing (equal). Daniel Paul: Formal analysis (equal); Investigation (equal); Software (equal); Visualization (equal); Writing – review & editing (equal). Patrick Wilhelm: Investigation (equal); Software (equal); Writing – review & editing (equal). Shaun G. Ard: Formal analysis (equal); Methodology (equal); Supervision (equal); Writing – review & editing (equal). Manfred Grieser: Investigation (equal); Writing – review & editing (equal). Robert von Hahn: Investigation (equal); Methodology (equal); Project administration (equal). Michael C. Heaven: Formal analysis (equal); Software (equal); Writing – review & editing (equal). Evangelos Miliordos: Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). Dominique Maffucci: Formal analysis (equal); Methodology (equal); Writing – review & editing (equal). Nicholas S. Shuman: Formal analysis (equal); Methodology (equal); Supervision (equal); Writing – review & editing (equal). Albert A. Viggiano: Conceptualization (equal); Funding acquisition (equal); Investigation (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal). Andreas Wolf: Conceptualization (equal); Investigation (equal); Project administration (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). Oldřich Novotný: Conceptualization (equal); Investigation (equal); Methodology (equal); Project administration (equal); Software (equal); Supervision (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

The Penning-sputtering ion source has been operated with O₂ and Ar gases and a Ti-dominated-alloy sputtering target. While the oxygen gas was isotopically pure ¹⁶O₂, for the used titanium target, a natural isotope distribution is expected, i.e., 8.3% of ⁴⁶Ti, 7.7% of ⁴⁷Ti, 73.7% of ⁴⁸Ti, 5.4% of ⁴⁹Ti, and 5.2% of ⁵⁰Ti. To maximize the yield of TiO⁺, the ⁴⁸Ti¹⁶O⁺ isotopologue with mass m = 64 u was chosen. Given the electrostatic acceleration scheme and the momentum filtering of the magnetic deflectors in the injection beamline, other ions of the same mass-to-charge ratio m/q = 64 u/e would get stored in the CSR along with ⁴⁸Ti¹⁶O⁺. To determine possible contaminant ions, the mass selected primary ions in the injection beamline were fragmented by collisions in an Ar-gas-cell, and the fragments were further mass analyzed by using a second magnetic filter.

Two isobaric contaminants have been considered in the analysis of the fragment mass spectra, i.e., $O4+16$ and ⁴⁷Ti¹⁶OH⁺. For $O4+16$ fragmentation, the detection of an O₂ fragment within the injection beamline could be used as a marker. Unfortunately, the corresponding m/q = 32 u/e fragment mass-to-charge ratio is masked by the doubly charged ⁴⁸Ti¹⁶O⁺⁺ molecular ion. To resolve the ambiguity, we also analyzed fragment mass spectra for primary ions of m/q = 63 and 62 u/e, corresponding to ⁴⁷Ti¹⁶O⁺ and ⁴⁶Ti¹⁶O⁺, respectively. Although for these primary-ion masses no contamination by $O4+$ is possible, the doubly charged ⁴⁷Ti¹⁶O⁺⁺ and ⁴⁶Ti¹⁶O⁺⁺ have been observed again. Moreover, the abundances of the respective ^ATi¹⁶O⁺⁺ isotopologues (A being the nucleon number of Ti) scaled well with the primary beam intensity and with other unique titanium markers, such as ^ATi⁺⁺⁺. From this behavior, we estimate that the $O4+16$ contamination did not exceed 1% of the total ion beam intensity. Furthermore, the DR imaging analysis did not show any fragmentation pattern, suggesting $O4+16$ contamination.

The potential ⁴⁷Ti¹⁶OH⁺ contamination was investigated by ⁴⁷Ti fragments and by corresponding isotopologues, i.e., ^A−1Ti fragments for ^A−1Ti¹⁶OH⁺ and ^ATi¹⁶O⁺ primary ions. For primary masses m/q = 64 and 63 u/e (A = 48 and 47, respectively), the ^A−1Ti fragments were clearly visible and we are unaware of any other contaminant at the corresponding mass. This was verified by choosing primary mass m/q = 62 u/e (A = 46) where the ^A−1Ti isotope does not exist, and we indeed did not observe any contribution at A − 1 = 45 in the fragment mass spectra. We estimate the relative fraction of ⁴⁷Ti¹⁶OH⁺ in the m/q = 64 u/e primary beam to be less than 10%, assuming that the fragmentation cross section for TiOH⁺ in the gas cell is at least as large as that for TiO⁺. As discussed further in  Appendix C 2 and Sec. IV B, using a specific fragmentation pattern in DR of TiOH⁺, we could further characterize the TiOH⁺ contribution to the imaging and rate data, respectively.

The TiO⁺ ions were produced in a hot sputtering Penning ion source, and excitation of their internal degrees of freedom is thus expected to reach several thousand kelvin.⁴³ After $∼40$ μs, the extracted ions reached the CSR, where they were first stored for a predefined amount of time to deexcite before the DR probing by electrons started. The ion cooling is dominated by radiative cascading to reach the equilibrium with the CSR radiation field, which can be approximated by two components:⁴⁴ 99% corresponding to $∼6$ K (representing the cryogenic chamber temperature) and 1% corresponding to 300 K (representing radiation leaks from outer environments, e.g., through the injection beamline). In this environment, we model the radiative relaxation of TiO⁺ using the Einstein coefficients for radiative transitions between the various electronic, vibrational, and rotational levels. As a result, we obtain TiO⁺ internal excitation distributions for the storage times equivalent to the DR probing in the experiment. Similar models of molecular-ion radiative cooling in the CSR radiation field were successfully tested in the past against in situ rotational state population probing methods.^44–46 

To the best of our knowledge, there are no published data on radiative lifetimes for transitions in TiO⁺. Therefore, we either estimate the lifetimes from other similar molecular systems or we calculate the corresponding Einstein coefficients for spontaneous emission ourselves. The radiative lifetimes for electronic transitions in the neutral TiO molecule⁴⁷ do not exceed 5 µs, which is similar also for other diatomic systems.⁴⁸ Therefore, in our cooling model, we neglect the higher lying TiO⁺ electronic states A²Σ⁺, B²Π, and a⁴Δ as they are expected to relax already during the travel from the ion source to the CSR.

To estimate the vibrational level lifetimes, we follow the approach of Amitay et al. (Ref. 49) for calculating rovibrational radiative lifetimes, while intentionally neglecting the fine-structure splitting of the ground state doublet TiO⁺(X²Δ). The level spacing was obtained from the spectroscopic TiO⁺ parameters as given in Sec. II A. The dipole function was calculated as μ = 6.3 D and dμ/dR = 6.2 × 10⁸ D/cm using a def2-TZVP basis set.⁵⁰ The vibrational radiative lifetime calculation reveals that even the slowest transition v = 1 ⟶ 0 does not exceed 60 ms. The vibrationally excited TiO⁺ levels are thus depopulated in CSR quickly, long before they could affect the DR probing phase, and therefore can be safely omitted from further discussion on the TiO⁺ radiative cooling model.

Thus, for the CSR experimental conditions, only rotational and TiO⁺(X²Δ_3/2;5/2) fine structure transitions remain relevant in the TiO⁺ radiative cooling model. For the remaining states, we again base the calculation of the Einstein coefficients for spontaneous emission on the approach of Amitay et al. (Ref. 49), but extend it for the treatment on multiplet electronic states.^51,52 We calculate the needed Hönl–London coefficients by considering the ²Δ state of TiO⁺ as intermediate between Hund’s coupling cases (a) and (b).⁵¹ For transitions within the Ω-branches, the results are nearly identical to pure Hund’s case (a),⁵¹ while the mixed character of the rotational wavefunctions gives rise to a non-negligible transition probability between the Ω-branches. The sum of the electric dipole Einstein coefficients from a single upper Ω = 5/2 level to all possible lower Ω = 3/2 levels is only weakly dependent on the J value of the upper level, and the related Ω = 5/2 ⟶ 3/2 radiative lifetime is $<700$ s for all levels. We also examined the possible role of magnetic dipole transitions and found that their additional contribution to the transition rates is ∼30% of those from the electric dipole.

We have generated a radiative relaxation model using the Einstein coefficients for spontaneous emission, while also accounting for stimulated emission and absorption by the CSR blackbody radiation. For the initial rotational excitation, we have taken a Boltzmann distribution at a temperature of 3000 K. The resulting rotational populations for the Ω = 3/2 and 5/2 branches of the electronic ground state are given in Fig. 2(b) as averages over four different storage time windows. It can be seen that, even though the initial rotational population spans widely over splitting of the Ω branches, at later storage times the two branches separate due to fast rotational cooling within each branch. The result of the radiative cooling model for TiO⁺ is summarized in Table III, listing the population fraction p_3/2 of the Ω = 3/2 branch and mean excitation energies $⟨ETiO+⟩$ averaged over various storage time windows. The uncertainties of the modeled populations were estimated from the systematic uncertainty of the employed dipole moment μ and initial internal temperature of the ions T_ini using extreme values of μ = 5.8–6.8 D and T_ini = 1000–5000 K as guide. We note that the radiative cooling model does not include additional state-changing effects, such as reactive depletion of states with high DR cross section and inelastic electron–ion collisions. While both processes may occur during the DR-probing phase of the measurement, we estimate that especially at long storage times the populations are dominantly given by the radiative cooling.

For TiO⁺ DR, a large number of densely-spaced E_KER channels can potentially contribute to the fragment imaging data. This is caused by the unknown excitation populations of Ti and O products of TiO⁺ DR (see Table I) and by the remaining fine-structure and rotational excitation of TiO⁺ even at long storage times (see  Appendix B). Additionally, due to the experiment-specific effects, the electron–ion collision energies follow a non-trivial, continuous distribution function f(E_c; E_d) even at fixed experimental conditions (fixed detuning energy E_d). To take these ambiguities in the energy balance equation [Eq. (5)] into account, we build a model transverse distance distribution f_⊥(d; ΔE) combining all the E_KER channels available at given reaction energy ΔE, while the individual E_KER sub-channels are still represented by the $f⊥0(d;EKER)$ distance distributions, as defined in Sec. III B. The individual channels and their weighting follow these rules:

• All available combinations of Ti and O product excitations (E_Ti + E_O < 1 eV) are considered as independent channels with individual DR cross sections and thus also independent weighting factors.

• The fine structure excitation branches Ω = 3/2 and 5/2 are also considered as independent channels with individual weighting factors.

• The $AITi;IO;Ω$ factors represent the weights for the I_Ti-th Ti-product channel, the I_O-th O-product channel, and the Ω fine structure branch. The corresponding combined excitation-balance energies E_Ω − E_Ti − E_O are listed in Table IV. Here, E_Ω is the fine-structure excitation energy of TiO⁺(X²Δ_Ω) with respect to the ground state.

• Rotational states J within one Ω branch follow populations p_J;Ω resulting from the radiative cooling model ( Appendix B).

• The collision energy distribution f_c(E_c; E_d) is considered as a continuous function based on the electron beam temperatures T_⊥ and T_‖ and other electron beam parameters (see  Appendix E and parameters in Sec. IV B).

• Background not originating from DR of TiO⁺ is represented by an additional distribution function $f⊥bg(d)$ with an independent scaling factor B.

We note that the model as given by Eq. (C1) assumes equal DR cross sections for rotational levels within the same Ω branch. While this assumption may generally not be correct (see, e.g., DR of HeH⁺, Ref. 25), treating the individual J-channels independently would make the fitting of the model f_⊥(d) to the data numerically unstable. The validity of neglecting the J dependence is also supported by our observation that the fitted value of ΔE is not too sensitive on varying the scaling of J-channels within the Ω-branch, i.e., on the rotational excitation distribution.

To describe the TiOH⁺ contribution in terms $f⊥bg(d)$ of the model distribution f_⊥(d) [Eq. (C1)], we first analyze the expected ranges of transverse fragment distances from TiO⁺ DR, based on the approximate reaction energy balance, and then compare it to the acquired data. For the relaxed ions at storage times $>500$ s, the TiO⁺ excitation energies stay below $ETiO+≲35$ meV (Table III). Thus, from Eq. (5) and from the lower estimate on the reaction energy ΔE ≳ −20 meV (Sec. II B), we derive that the relative product kinetic energies do not exceed E_KER ≈ 45 meV at E_d = 0 eV and stay below E_KER ≈ 200 meV even for E_d = 150 meV. Correspondingly, at these detuning energies, the transverse distances do not exceed d = 4 and 8 mm, respectively. However, the acquired $f⊥̃(d)$ distributions for cold ions at E_d = 0 eV (Fig. 4, $>500$ s) display clear contribution at d > 4 mm. This tail displays similar behavior also at non-zero collision energies, i.e., the main component smoothly decays from d ≈ 4 mm up to d ≈ 22 mm, and then further extends as an even weaker, nearly constant contribution until the detector size limit. As these distances cannot originate from DR of TiO⁺, we assign it to DR of TiOH⁺.

In the model distance distribution f_⊥(d) [Eq. (C1)], the TiOH⁺ DR is represented by the term $Bf⊥bg(d)$⁠, where $f⊥bg(d)$ is the same for all fitted data (various E_d and storage times), while the scaling parameter B is determined in each fit. The $f⊥bg(d)$ distribution is approximated by a linear function, which well represents the data at d = 8–15 mm. The real shape of $f⊥bg(d)$ at low distances with overlapping DR of TiO⁺ (especially for d < 4 mm) is unknown. However, it can be well expected that with decreasing d, the distribution has to eventually reach also zero amplitude, similarly as the $f⊥0(d)$ distribution for single E_KER. Various shapes of $f⊥bg(d)$ in this region have been tested, and only negligible effects on the fitted ΔE result were found.

The TiOH⁺ contribution to the rate coefficient data is further discussed in Sec. IV B.

Finally, N_NDF(E_d) is the number of degrees of freedom relevant for the fit in the given E_d dataset, i.e., the difference between the number of fitted experimental datapoints and the number of free fit parameters. Both quantities entering N_NDF(E_d) vary as a function of E_d for two reasons: (i) The upper limit of the fragment distance fit range (and thus also the number of points) was kept as low as possible for each E_d so that mainly distances covered by available TiO⁺ channels were covered. Moreover, (ii) only energetically open channels from Table IV at E_c ≈ E_d for the given E_d were included, i.e., E_KER > 0 (keeping a fixed list of energetically open channels for each E_d dataset in all the fits for the various ΔE). Additionally, for numerical stability reasons, we merged channels with combined excitation-balance energies E_Ω − E_Ti − E_O lying close to each other. By applying an energy difference limit of $<3.2$ meV, the number of channels thus reduced from 18 to 9, and the number of free amplitude parameters $AITi;IO;Ω$ reduced correspondingly. The resulting groups are indicated in Table IV by horizontal separators. For each such group of channels, the combined excitation-balance energy was replaced by the mean value from the contributing channels. The applied channel grouping is not expected to change the fit results beyond other systematic uncertainties entering the model. With these modifications, N_NDF(E_d) ranged from 16 to 52.

Since the DR rate coefficient peaks at E_d = 0 eV, the corresponding dataset was acquired with much higher statistical quality compared to the data from higher detuning energies, which can also be seen by comparing the left and right columns in Fig. 5. To reduce the resulting strong weight of the E_d = 0 eV dataset in the $χred2(ΔE)$ function, we deliberately increased the statistical uncertainties σ_i for the E_d = 0 eV data by a factor of 4 before the summation in Eq. (C4).

In Sec. IV A, we discuss the usage of $χred2(ΔE)$ to determine the experimental reaction energy ΔE. We also employ the quantity $χred,Ed2(ΔE)=χ2(ΔE,Ed)/NNDF(Ed)$ representing the reduced χ² obtained for only one selected E_d dataset. Figure 12 shows an example of the model transverse distance distributions fitted to the experimental data at the 500–600 s storage time window and various detuning energies E_d.